The Mean-Value Theorem

The Mean Value Theorem is one of the most important
theoretical tools in Calculus. It states that if f(x) is
defined and continuous on the interval [a,b] and differentiable
on (a,b), then there is at least one number c in the interval
(a,b) (that is a < c < b) such that

The special case, when
f(a) = f(b) is known as Rolle's
Theorem. In this case, we have f '(c) =0. In other words,
there exists a point in the interval (a,b) which has a
horizontal tangent. In fact, the Mean Value Theorem can be stated
also in terms of slopes. Indeed, the number

is the slope of the line passing through (a,f(a)) and
(b,f(b)). So the conclusion of the Mean Value Theorem states
that there exists a point
such that the tangent line
is parallel to the line passing through (a,f(a)) and
(b,f(b)). (see Picture)

Example. Let
,
a = -1and b=1. We have

On the other hand, for any
,
not equal to 0, we have

So the equation

does not have a solution in c. This does not
contradict the Mean Value Theorem, since f(x) is not even
continuous on [-1,1].

Remark. It is clear that the derivative of a constant
function is 0. But you may wonder whether a function with
derivative zero is constant. The answer is yes. Indeed, let
f(x) be a differentiable function on an interval I, with
f '(x) =0, for every .
Then for any a and b in
I, the Mean Value Theorem implies

for some c between a and b. So our assumption implies

Thus
f(b) = f(a) for any aand b in I, which means that f(x) is constant.